Variational Quantum Operator Simulation

Abstract

Implementing time-evolution operators in shallow quantum circuits is important for quantum simulations. The standard method of Trotterization requires a large number of gates to achieve practical accuracy. Variational Quantum Simulation (VQS) is an algorithm that calculates the time evolution of a quantum state and can be executed with shallower circuits than Trotterization. However, the operator obtained by VQS evolves only a fixed initial state and is not the time evolution operator itself. In this paper, we propose Variational Quantum Operator Simulation (VQOS), a method to realize time evolution operators in shallow quantum circuits. This method is based on the variational principle for operators and does not require the implementation of the desired Trotter decomposition of the time evolution operator. We performed numerical simulations of the VQOS algorithm and successfully implemented the time evolution operator for closed systems in a quantum circuit that is up to 5 times shallower than the Trotterization. By providing a more practical way to implement time evolution operators, VQOS increases the applicability of near-term quantum computers.

Figures (7)

• Figure 1: Circuits to estimate Eq. \ref{['qe:vqs_gjkl']}. (a) Indirect measurement method. $g_{jkl}$ is estimated by measuring ancilla qubit $\langle Z\rangle_{j,k,l}$. (b) Direct measurement method. $\mathcal{M}_{P_j}$ is the measurement with Pauli operator $P_j$. $g_{jkl}$ is estimated as $p(M_{P_j}=+1)\langle P_k\rangle_{M_{P_j}=+1} - p(M_{P_j}=-1)\langle P_k\rangle_{M_{P_j}=-1}$. $p(M_{P_j}=\pm1)$ is the probability that the outcome of the mid-measurement is $\pm1$. $\langle P_k\rangle_{M_{P_j}=\pm1}$ is the expectation value of $P_k$ conditional on $M_{P_j}=\pm1$.
• Figure 2: Quantum circuits for estimating Eqs. \ref{['eq:N_imp']} and \ref{['eq:W_imp']}. (a) A simplified, resource-reduced implementation of VQOS obtained by tracing out the idling register in (a'). The expectation value of the measurement result of the ancilla qubit is $g_{jkl}$. (a') The circuit implementation of VQOS obtained by directly performing VQSS on the Choi state with indirect measurement. (b) A simplified, resource-reduced implementation of VQOS obtained by tracing out the idling register in (b') and replacing mid-circuit measurement with random initialization to eigenstates of $P_j$ belonging to the eigenvalues $+1~(-1)$. (b') The circuit implementation of VQOS obtained by directly performing VQSS on the Choi state with direct measurement. $\mathcal{M}_{P_j}$ represents the measurement of $P_j$. $\ev{P_k}$ is measured at the end of the circuit.
• Figure 3: One layer of variational circuit $U(\bm\theta)$ when $n=4$, consisting of Rx gates, Rxx gates, Ryy gates, and Rzz gates, with the same coupling as the Hamiltonian. The Rx gate is defined as $\exp(-i X \theta/2)$ and Rxx gate as $\exp(-i X \otimes X \theta/2)$, and the same for Ryy and Rzz gates. Each rotation gate has different parameters.
• Figure 4: The process infidelity $1-f$ of the time evolution operators of the $9$-site 1D periodic boundary transverse the magnetic field Heisenberg model obtained by VQOS and Trotterization against the exact operator obtained by diagonalization of the Hamiltonian; the solid line is the result of VQOS, and the dotted line is the result of Trotterization. The blue, orange, and green plots correspond to 10, 20, and 30 layers, respectively.
• Figure 5: Relationship between number of layers and operator infidelity $1-f$ at $t=1$, $3$, and $5$ in VQOS and Trotterization. Results for different sites correspond to different colored plots. The plots with round and square markers correspond to the results of VQOS and Trotterization, respectively.